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Encyclopedia of Microtonal Music Theory

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temperament

[Joe Monzo]

A musical tuning which is not just-intonation; that is, the intervals are not small-integer ratios.

The most familiar types of temperament today are the equal temperaments, but historically many temperaments have also been unequal; all well-temperaments, and all those belonging to the meantone family, fall into the latter category.

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[Graham Breed, Yahoo tuning, message 8519 (Sun Feb 13, 2000 6:17 am)]

I would prefer:

A tuning which approximates just-intonation; that is, the intervals are close to small-integer ratios, and the scale is somehow simplified relative to its just equivalent.

Some tunings make no attempt to approximate JI, and so should not be considered temperaments.

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[John Chalmers, Divisions of the Tetrachord]

A tuning system which contains tempered intervals.

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[Gene Ward Smith, Yahoo tuning, message 55540 (Sat Aug 14, 2004 5:14 pm)]

I would say a regular temperament is determined by a homomorphic mapping from the p-limit, or possibly another finitely-generated subgroup of the positive rationals, to an abstract free group of smaller rank. This can be specified (uniquely determined) by giving a wedgie, a kernel, or an explicit mapping.

Note I do not include the tuning map as part of the definition, so this is an abstract definition of what a regular temperament is. However, this is how the word is most commonly used; people may object to it but the same people talk of 1/4-comma meantone or 2/7-comma meantone as if they were both meantone. My definition also says that even though 31-et meantone is tuned to a group of rank one, it is still qua meantone a group of rank two, and the tuning mapping is another issue.

You might also note that this definition, which says a temperament is a morphism, makes no sense unless you get the category right, which connects to the thread about spaces.

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